Supersonic Wind Tunnel Part 2 – Initial Nozzle Design

The most critical part of the wind tunnel is the nozzle, it must accelerate the flow to the desired Mach number while leaving the test section free of shock waves. The two methods I have used have been a numerical technique to create the geometry and later on I’ll use computational fluid dynamics (CFD) to see how the section performs.

The numerical method is known as Characteristics and is detailed nicely in Anderson’s

Characteristic curves between three nodes

book ‘Modern Compressible flow’ but I’ll go over the basics to explain it as best I can. It’s based on the assumption that two-dimensional, steady, irrotational flow can be expressed as a non-linear differential equation of velocity potential. The properties of Characteristics can be defined by the following, that in two-dimensional supersonic flow, a characteristic is a curve along which flow properties are continuous and propagate at the local speed of sound.

The image above shows two points ab connected to a third point d, the line connecting d to a, is defined as a ‘right running characteristic’ and db is a left running characteristic. This means that starting from a line of points, you then find the properties and locations of subsequent points downstream, creating a ‘net’ as you go. This pattern of expansion waves for a nozzle, like the simple example shown here, agrees with the pattern of shockwaves often seen in Schlieren imaging.

Characteristic net in a nozzle

The method is relatively straight forward to compute, and can reduced to a set of simple equations. Using the simple example above, along the curve ac where the initial velocity and flow angles are known (this is the starting curve or line). The initial speed will be Mach 1 as the flow is choked in the throat, the flow angles are simply found by interpolating between 0° along the centreline and  at the wall (the divergence angle). These values are represented as the Prandtl-Meyer angle  and the flow angle . Points upstream, such as point b, are found using the following:


This is then repeated for points bc, de…. and so on, creating the net.

The Prandtl-Meyer function is shown below, and trying to solve for the Mach number leads you to an equation that cannot be solved explicitly for M, and a solution must either be found through iteration or through the approximation given by [1]. I found an easier way by using the built in function within Matlab, [Prandtl-Meyer Solver Matlab].


I’m an engineer, if there’s an easy way to do things I do it, at least it saves writing a few more lines of code, that effort could be used to make another cup of coffee at least!

Once the Mach number is known, the Mach angle can be found using the standard equation:


The x coordinates of each point n, can then be found using the following:


where the coordinates of the previous points are used, with the gradients of the lines leading to the new point, m;


So having spent a couple of days in the run up to Christmas writing all this up into a Matlab script, and have been exploring various options. I have settled for a nozzle that should now produce the required Mach number of 2.5 in the test section, while keeping the that test section free of shock waves. I will be spending the next couple of days generating a mesh of the nozzle geometry and using ANSYS Fluent to run some CFD simulations to verify the design. My background is the use of CFD to analyse ship designs, so this shouldn’t pose too much difficulty, I’ll update with how I did this soon. But for now here are some results of the Characteristic analysis showing the expansion waves and a contour plot of the Mach number within the nozzle itself.

Nozzle Wave Data

The expansion wave analysis and Mach contours for two slightly different Mach 2.5 nozzle profiles.

I still don’t know how to approach the actual building of the nozzle, I have thought of 3D printing, then quickly discounted it as the surface is less than perfect even with an acetone wash, which then can warp the geometry, by how much and what effect that would have I don’t know. I’m far too poor to machine this in aluminium, which would be my optimal choice, so the two methods I still haven’t settled on is the use of MDF board, which can then be soaked with epoxy and smoothed to a very high finish or casting in epoxy resin with either  chopped glass fibre mat or microballoons for added strength. The latter option would allow me to cast captive nuts inside to make the attachment of the Plexiglas side panels easier, as well as the ability to cast grooves to allow o-ring cord to give better sealing than a gasket alone would allow.

I also started giving thought to a traverse system to move things like test models and Pitot tubes within the test section, without having to manually adjust them. I acquired a couple of old flatbed scanners to salvage the stepper motors and tracks. I’m thinking of creating a dashboard in LabView that will allow me to control all aspects of the tunnel from my computer, from opening the valve, monitoring internal pressure of the tank, pressure along the tunnel walls, temperatures, the lift and drag of the test model on the sting…. I did something similar once for small gas turbines that worked well, plus with the short run times having the tunnel valve operation, data recording and other things done autonomously is probably essential.


[1] Ralph Carmichael,

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